Hierarchical Prisoner's Dilemma in Hierarchical Public-Goods Game

TL;DR

This paper introduces a hierarchical public-goods game model revealing a multi-level prisoner's dilemma, where individual and group incentives conflict, and explores conditions under which cooperation or defection emerges at different levels.

Contribution

The study formulates a hierarchical public-goods game capturing multi-level strategic interactions and uncovers the existence of an optimal group size and conditions for resolving the hierarchical prisoner's dilemma.

Findings

Groups tend to choose defection strategies like armaments as Nash equilibrium.

An optimal group size exists that maximizes group payoff.

Large asymmetries in population lead smaller groups to cooperate to avoid conflict.

Abstract

The dilemma in cooperation is one of the major concerns in game theory. In a public-goods game, each individual pays a cost for cooperation, or to prevent defection, and receives a reward from the collected cost in a group. Thus, defection is beneficial for each individual, while cooperation is beneficial for the group. Now, groups (say, countries) consisting of individual players also play games. To study such a multi-level game, we introduce a hierarchical public-goods (HPG) game in which two groups compete for finite resources by utilizing costs collected from individuals in each group. Analyzing this HPG game, we found a hierarchical prisoner's dilemma, in which groups choose the defection policy (say, armaments) as a Nash strategy to optimize each group's benefit, while cooperation optimizes the total benefit. On the other hand, for each individual within a group, refusing to pay the cost (say, tax) is a Nash strategy, which turns to be a cooperation policy for the group, thus leading to a hierarchical dilemma. Here, the reward received by one group increases with the population, as does the collected cost. In spite of this, we find that there exists an optimal group size that maximizes its payoff. Furthermore, when the population asymmetry between two groups is large, a smaller group will choose a cooperation policy (say, disarmament) to avoid excessive response from the larger group, which leads to the resolution of the prisoner's dilemma between the groups. The relevance of the HPG game to policy selection in society and the optimal size in human or animal groups are discussed accordingly.